#8127460

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,15,17)(11,20,23,16)(12,24,19,14)(13,22,21,18)(26,30)(28,35)(31,37)(33,36);(1,2,4,6)(3,5,8,7)(9,11,15,23)(10,16,17,20)(12,13,19,21)(14,22,24,18)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40);(1,2)(3,5)(4,6)(7,8)(10,17)(12,19)(13,21)(16,20)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39);(1,3,4,8)(2,5,6,7)(9,12,15,19)(10,14,17,24)(11,21,23,13)(16,18,20,22)(25,28,29,35)(26,32,30,38)(27,33,34,36)(31,39,37,40);(9,13,15,21)(10,18,17,22)(11,19,23,12)(14,20,24,16)(25,27)(26,31)(28,33)(29,34)(30,37)(32,39)(35,36)(38,40);(9,14)(10,19)(11,22)(12,17)(13,20)(15,24)(16,21)(18,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034656);